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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '736.44', 'ambient_counter': 44, 'ambient_order': 736, 'ambient_tex': 'C_{46}.C_4^2', 'central': True, 'central_factor': False, 'centralizer_order': 736, 'characteristic': True, 'core_order': 23, 'counter': 39, 'cyclic': True, 'direct': True, 'hall': 23, 'label': '736.44.32.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '32.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '32.2', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 32, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2.C_4^2', 'simple': True, 'solvable': True, 'special_labels': ['C5'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '23.1', 'subgroup_hash': 1, 'subgroup_order': 23, 'subgroup_tex': 'C_{23}', 'supersolvable': True, 'sylow': 23}
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gps_subgroup_data • Show schema
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{'ambient': '736.44', 'aut_centralizer_order': 384, 'aut_label': '32.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '22.2', 'aut_weyl_index': 384, 'centralizer': '1.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['16.a1.a1', '16.b1.a1', '16.b1.b1', '16.b1.c1', '16.c1.a1', '16.c1.b1', '16.c1.c1'], 'contains': ['736.a1.a1'], 'core': '32.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [8657, 3526, 3654, 6560, 7159, 7585, 4032, 3967], 'generators': [32], 'label': '736.44.32.a1.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '32.a1.a1', 'normal_contained_in': ['16.b1.a1', '16.b1.c1', '16.b1.b1', '16.c1.c1', '16.c1.a1', '16.c1.b1', '16.a1.a1'], 'normal_contains': ['736.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '32.a1.a1', 'projective_image': '32.2', 'quotient_action_image': '1.1', 'quotient_action_kernel': '32.2', 'quotient_action_kernel_order': 32, 'quotient_fusion': None, 'short_label': '32.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '23.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 22, 'aut_gen_orders': [22], 'aut_gens': [[1], [14]], 'aut_group': '22.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 22, 'aut_permdeg': 13, 'aut_perms': [515289600], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [23, 1, 22, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{22}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 22, 'autcent_group': '22.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 22, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{22}', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [23, 1, 22]], 'center_label': '23.1', 'center_order': 23, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['23.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [23, 1, 22, 1]], 'element_repr_type': 'PC', 'elementary': 23, 'eulerian_function': 1, 'exponent': 23, 'exponents_of_order': [1], 'factors_of_aut_order': [2, 11], 'factors_of_order': [23], 'faithful_reps': [[1, 0, 22]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '23.1', 'hash': 1, 'hyperelementary': 23, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 22, 'irrQ_dim': 22, 'irrR_degree': 2, 'irrep_stats': [[1, 23]], 'label': '23.1', 'linC_count': 22, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 22, 'linQ_degree_count': 1, 'linQ_dim': 22, 'linQ_dim_count': 1, 'linR_count': 11, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C23', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 23, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 23, 'order_factorization_type': 1, 'order_stats': [[1, 1], [23, 22]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 22, 'outer_gen_orders': [22], 'outer_gen_pows': [0], 'outer_gens': [[14]], 'outer_group': '22.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 22, 'outer_permdeg': 13, 'outer_perms': [515289600], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{22}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 23, 'pgroup': 23, 'primary_abelian_invariants': [23], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [22, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -23]}, 'Lie': [{'d': 1, 'q': 23, 'gens': [1177652997443428940313], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 23, 'gens': [12191]}, 'Perm': {'d': 23, 'gens': [24728016011107368960000]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [23], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{23}', 'transitive_degree': 23, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '368.19', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 132, 'aut_gen_orders': [44, 44, 66, 44], 'aut_gens': [[1, 4, 16], [5, 382, 432], [5, 6, 560], [375, 1, 592], [373, 374, 48]], 'aut_group': None, 'aut_hash': 279789898593680170, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 8448, 'aut_permdeg': 46, 'aut_perms': [706697520454960613305821277467022760534841001211664889380, 654017855850272889266530757679205176092587276536891884528, 3896857858072960055041900683540236627903123735261376924804, 2617879710170104518346739019414114229309555958882709059823], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 3, 2], [4, 2, 12, 1], [23, 1, 22, 1], [46, 1, 22, 1], [46, 1, 66, 2], [92, 2, 264, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{22}\\times C_2^4:C_3.D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 22, 'autcent_group': '1408.2328', 'autcent_hash': 2328, 'autcent_nilpotent': True, 'autcent_order': 1408, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6\times C_{22}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 7], [4, 2, 12], [23, 1, 22], [46, 1, 154], [92, 2, 264]], 'center_label': '184.12', 'center_order': 184, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '23.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 44, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['23.1', 1], ['32.2', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [4, 2, 2, 6], [23, 1, 22, 1], [46, 1, 22, 7], [92, 2, 44, 6]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 24, 'exponent': 92, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3, 11], 'factors_of_order': [2, 23], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '92.4', 'hash': 44, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 1], 'inner_gens': [[1, 372, 16], [369, 4, 16], [1, 4, 16]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 368], [2, 92]], 'label': '736.44', 'linC_count': 2335872, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 28, 'linQ_degree_count': 304, 'linQ_dim': 28, 'linQ_dim_count': 196, 'linR_count': 19272, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C46.C4^2', 'ngens': 6, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 460, 'number_divisions': 28, 'number_normal_subgroups': 52, 'number_subgroup_autclasses': 26, 'number_subgroup_classes': 76, 'number_subgroups': 100, 'old_label': None, 'order': 736, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 7], [4, 24], [23, 22], [46, 154], [92, 528]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 132, 'outer_gen_orders': [22, 44, 44, 44], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[382, 379, 48], [13, 374, 272], [380, 379, 144], [379, 7, 304]], 'outer_group': None, 'outer_hash': 5293298213448331449, 'outer_nilpotent': False, 'outer_order': 2112, 'outer_permdeg': 21, 'outer_perms': [12806637000918526719, 17929136188035158166, 24329020081766486009, 10245309949447781455], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_{22}\\times \\GL(2,\\mathbb{Z}/4)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 35, 'pgroup': 0, 'primary_abelian_invariants': [4, 4, 23], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [22, 4], [44, 10]], 'representations': {'PC': {'code': 1611635419925416195745621, 'gens': [1, 3, 5], 'pres': [6, -2, -2, -2, -2, -2, -23, 12, 6698, 50, 88]}, 'Perm': {'d': 35, 'gens': [304179247495769143992875691917967360000, 274399914062452002790228623360000, 24728016011107368960000, 837605342339873243136000000, 607832233317669660028613481332736000000, 548189243611861521181704192000000]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 92], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{46}.C_4^2', 'transitive_degree': 736, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 3, 2, 2, 2, 2], 'aut_gens': [[1, 4, 8], [29, 4, 24], [21, 4, 26], [7, 4, 28], [25, 4, 23], [19, 4, 28], [7, 4, 26], [5, 4, 12], [5, 4, 8]], 'aut_group': '384.20100', 'aut_hash': 20100, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 384, 'aut_permdeg': 12, 'aut_perms': [235050896, 424023308, 183902764, 207622653, 140104852, 177005226, 165757252, 182450408], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 3, 2], [4, 2, 12, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\wr S_3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 7], [4, 2, 12]], 'center_label': '8.5', 'center_order': 8, 'central_product': False, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [4, 2, 2, 6]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 1, 2], 'inner_gens': [[1, 4, 12], [1, 4, 8], [5, 4, 8]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 4]], 'label': '32.2', 'linC_count': 192, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 76, 'linQ_dim': 6, 'linQ_dim_count': 49, 'linR_count': 49, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2.C4^2', 'ngens': 2, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 20, 'number_divisions': 14, 'number_normal_subgroups': 26, 'number_subgroup_autclasses': 13, 'number_subgroup_classes': 38, 'number_subgroups': 50, 'old_label': None, 'order': 32, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7], [4, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 6, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[3, 4, 9], [3, 4, 24], [30, 4, 25], [17, 4, 10], [7, 4, 30]], 'outer_group': '96.195', 'outer_hash': 195, 'outer_nilpotent': False, 'outer_order': 96, 'outer_permdeg': 8, 'outer_perms': [127, 23, 1442, 11520, 5160], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,\\mathbb{Z}/4)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 12, 'pgroup': 2, 'primary_abelian_invariants': [4, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10]], 'representations': {'PC': {'code': 67764297, 'gens': [1, 3, 4], 'pres': [5, 2, 2, 2, 2, 2, 10, 243, 58]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101743012388990, 91655865498367938]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [122109387504, 122297461254, 30594431879, 149885733128, 61113682502]}, 'Perm': {'d': 12, 'gens': [43908489, 5887, 16, 87091216, 11520]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 4], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2.C_4^2', 'transitive_degree': 32, 'wreath_data': None, 'wreath_product': False}
